We study training one-hidden-layer ReLU networks in the neural tangent kernel (NTK) regime, where the networks' biases are initialized to some constant rather than zero. We prove that under such initialization, the neural network will have sparse activation throughout the entire training process, which enables fast training procedures via some sophisticated computational methods. With such initialization, we show that the neural networks possess a different limiting kernel which we call bias-generalized NTK, and we study various properties of the neural networks with this new kernel. We first characterize the gradient descent dynamics. In particular, we show that the network in this case can achieve as fast convergence as the dense network, as opposed to the previous work suggesting that the sparse networks converge slower. In addition, our result improves the previous required width to ensure convergence. Secondly, we study the networks' generalization: we show a width-sparsity dependence, which yields a sparsity-dependent Rademacher complexity and generalization bound. To our knowledge, this is the first sparsity-dependent generalization result via Rademacher complexity. Lastly, we study the smallest eigenvalue of this new kernel. We identify a data-dependent region where we can derive a much sharper lower bound on the NTK's smallest eigenvalue than the worst-case bound previously known. This can lead to improvement in the generalization bound.
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Analyzing Generalization of Neural Networks through Loss Path Kernels
Deep neural networks have been increasingly used in real-world applications,
making it critical to ensure their ability to adapt to new, unseen data. In this paper,
we study the generalization capability of neural networks trained with (stochastic)
gradient flow. We establish a new connection between the loss dynamics of
gradient flow and general kernel machines by proposing a new kernel, called
loss path kernel. This kernel measures the similarity between two data points by
evaluating the agreement between loss gradients along the path determined by the
gradient flow. Based on this connection, we derive a new generalization upper
bound that applies to general neural network architectures. This new bound is tight
and strongly correlated with the true generalization error. We apply our results to
guide the design of neural architecture search (NAS) and demonstrate favorable
performance compared with state-of-the-art NAS algorithms through numerical
experiments.
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- Award ID(s):
- 2107189
- PAR ID:
- 10518466
- Publisher / Repository:
- NeurIPS 2023
- Date Published:
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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